Power spectrum model of visual masking: simulations and empirical data.

Serrano-Pedraza et al.

Vol. 30, No. 6 / June 2013 / J. Opt. Soc. Am. A

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Power spectrum model of visual masking: simulations and empirical data Ignacio Serrano-Pedraza,1,2,* Vicente Sierra-Vázquez,1 and Andrew M. Derrington3 1

Departamento de Psicología Básica I, Facultad de Psicología, Universidad Complutense, Campus de Somosaguas, Madrid 28223, Spain 2 Institute of Neuroscience, Faculty of Medical Sciences, Newcastle University, Newcastle upon Tyne NE2 4HH, UK 3 Faculty of Humanities and Social Sciences, University of Liverpool, Liverpool L69 7ZX, UK *Corresponding author: [email protected] Received February 25, 2013; accepted April 14, 2013; posted April 16, 2013 (Doc. ID 185932); published May 14, 2013 In the study of the spatial characteristics of the visual channels, the power spectrum model of visual masking is one of the most widely used. When the task is to detect a signal masked by visual noise, this classical model assumes that the signal and the noise are previously processed by a bank of linear channels and that the power of the signal at threshold is proportional to the power of the noise passing through the visual channel that mediates detection. The model also assumes that this visual channel will have the highest ratio of signal power to noise power at its output. According to this, there are masking conditions where the highest signal-to-noise ratio (SNR) occurs in a channel centered in a spatial frequency different from the spatial frequency of the signal (off-frequency looking). Under these conditions the channel mediating detection could vary with the type of noise used in the masking experiment and this could affect the estimation of the shape and the bandwidth of the visual channels. It is generally believed that notched noise, white noise and double bandpass noise prevent off-frequency looking, and high-pass, low-pass and bandpass noises can promote it independently of the channel’s shape. In this study, by means of a procedure that finds the channel that maximizes the SNR at its output, we performed numerical simulations using the power spectrum model to study the characteristics of masking caused by six types of onedimensional noise (white, high-pass, low-pass, bandpass, notched, and double bandpass) for two types of channel’s shape (symmetric and asymmetric). Our simulations confirm that (1) high-pass, low-pass, and bandpass noises do not prevent the off-frequency looking, (2) white noise satisfactorily prevents the off-frequency looking independently of the shape and bandwidth of the visual channel, and interestingly we proved for the first time that (3) notched and double bandpass noises prevent off-frequency looking only when the noise cutoffs around the spatial frequency of the signal match the shape of the visual channel (symmetric or asymmetric) involved in the detection. In order to test the explanatory power of the model with empirical data, we performed six visual masking experiments. We show that this model, with only two free parameters, fits the empirical masking data with high precision. Finally, we provide equations of the power spectrum model for six masking noises used in the simulations and in the experiments. © 2013 Optical Society of America OCIS codes: (330.1800) Vision - contrast sensitivity; (330.4060) Vision modeling; (330.5510) Psychophysics. http://dx.doi.org/10.1364/JOSAA.30.001119

1. INTRODUCTION The existence of linear visual mechanisms or channels [1], each selectively sensitive to a limited range of spatial frequencies and acting in parallel, is widely accepted [2–6]. In order to study the spatial characteristics of these visual channels (i.e., shape and spatial-frequency bandwidth), the power spectrum model of masking, adapted from the study of auditory filters [7–9], is one of the most used in vision [10–18]. In addition, this model has also been used to interpret the results of masking experiments with tasks other than grating detection as letter identification in normal [19–24] and amblyopic observers [25]. According to this model, which comes originally from auditory psychophysics [7,26], the power of a sinusoidal waveform masked with noise at contrast detection threshold is proportional to the power of the noise passing through the channel that mediates detection [11,13,16]. Thus, the contrast detection threshold of a masked sinusoidal waveform can be used as an indirect measure of the area of overlap between the visual channel and the noise spectrum, and, as a consequence, some tuning characteristics of the channel can be inferred 1084-7529/13/061119-17$15.00/0

from the relationship between contrast detection threshold and noise parameters. Although the power spectrum model is simple and effective, other detection models, including the square-law detector [27], the energy detector [28], or the envelope detector model [27,28], have also been used to interpret results from auditory [29] and visual masking experiments [11,30]. To compare these models with the power spectrum model is the subject of a future paper. In the framework of the power spectrum model of visual masking, a variety of masking noises have been used to study the visual channels. These include white noise [16,17], highpass and low-pass noise [10,11,16,23,24,30], bandpass noise of fixed bandwidth [10–12], bandpass noise of increasing bandwidth [10,11], band-stop or “notched” noise [13,14,31], and double bandpass noise [12]. Although it is supposed that a bank of channels processes the visual inputs, the power spectrum model assumes that a single channel mediates detection; in other words, the channel with the highest signal-to-noise ratio (SNR) detects the signal. © 2013 Optical Society of America

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This means, for example, that when no noise is added to the stimulus, the single channel involved in detection is tuned to the spatial frequency of the signal. However, masking experiments have shown that the channel involved in detection may not be centered on the spatial frequency of the signal and can vary with the type of noise used in the masking experiment [11–13,16]. These experiments have shown that the observer can use a channel centred at a spatial frequency other than the signal’s spatial frequency in order to improve the SNR and the consequence is a reduction of the detection threshold. This detection strategy has been called “off-frequency looking” [11] and was first found in auditory psychophysics (called off-frequency listening by [32], and illustrated in [33], its figure 2). In the current paper we will use the same meaning of the phrase “off-frequency looking” as described by [8], (p. 158): “to refer to conditions in which the signal is detected through a filter that is not centered on the signal frequency.” Off-frequency looking is a problem because if we use the power spectrum model to analyze the masking data (i.e., using low-pass or high-pass noises) without taking into account the presence of off-frequency looking, and the subject used this strategy, then the model will not estimate correctly the shape and the bandwidths of the visual channels. One solution will be to use masking noises that prevent off-frequency looking. It is generally believed that notched noise [13,31,33], white noise (all-pass) [17], and double bandpass noise [12,34] prevent off-frequency looking and that low-pass, high-pass, and bandpass noises facilitate it [11]. Unfortunately, as far as we can tell, this general belief has never been tested: we have not found theoretical studies in which the power spectrum model has been properly tested with these types of noises in order to study the conditions in which off-frequency looking takes place. (As far as we know the only exception is the work of [35] in hearing but, due to the aim of their work, they used a special type of notched noise deliberately designed to promote off-frequency listening and a particular functional form for the gain function of auditory filters). The aim of this work is, by means of numerical simulations using the power spectrum model, to study the characteristics of masking caused by the six most frequently used types of noise (all-pass, low-pass, high-pass, bandpass of fixed bandwidth, notched, and double bandpass) when we make different assumptions about the symmetry of the shape and about the bandwidth of the channels. Contrary to the generally accepted idea, we will show that notched and double bandpass noises fail to prevent off-frequency looking except when these noises fulfill certain conditions and depending on the assumed (symmetric or asymmetric) shape of the channel involved in detection. Although the thresholds predicted by the model are very similar to empirical data, we tested this model using empirical masking data. We performed different visual masking experiments using the six types of one-dimensional (1D) noise that were used in the simulations. We fitted the power spectrum model to 41 contrast thresholds and our results show that with only two free parameters, the model fits the data with high precision showing that this classical model has excellent explanatory power. Finally, in order to facilitate the use of the power spectrum model of visual masking by others, we provide the definite integrals for the two proposed shapes of the channels, and


the expressions of the integrals for each type of noise (all-pass, low-pass, high-pass, notched, bandpass, and double bandpass noise) (see Appendix A).

2. POWER SPECTRUM MODEL OF VISUAL MASKING A. Model Description The power spectrum model of masking makes three main assumptions [36]: (i) the stimulus (signal in a noise background) is processed by a bank of separate, independent, and overlapping bandpass linear filters; (ii) the threshold for detecting the signal is determined solely by the amount of noise passing through the single filter involved in detection; and (iii) threshold is assumed to correspond to a constant SNR at the output of the filter. This model is referred to as the “power spectrum model” of masking because it only takes into account the longterm power spectrum of the stimulus. The possible effects on threshold of other factors, such as the relative phase of the components or the variability of the noise, are ignored [8,9]. In its visual version, the power spectrum model of visual masking predicts that the squared-contrast detection threshold of a sinusoidal grating (m2T u0 with spatial frequency u0 ) used as signal test and masked with visual 1D noise of power spectrum ρ, is m2T u0

m20 ξk 4s

R ∞

ρujHu; ξk j2 du ; jHu0 ; ξk j2 0

(1)

where u is the spatial frequency, in cycles per degree (c/deg), ξk is the peak spatial frequency of the channel that detects the signal with spatial frequency u0 , m0 ξk corresponds to the contrast detection threshold of a grating of spatial frequency ξk without external noise, s is a sensitivity parameter [16], and jHu; ξk j is the modulation transfer function (MTF) of the channel centered at the spatial frequency ξk . (Note: jHu; ξk j2 is the attenuation characteristic or gain function of channel centered at ξk .) Equation (1) was derived in the vision literature by [16] (see his Eqs. (4) and (5), with minor differences in notation, see also a deduction of this equation in [37], pp. 34–37). The version of Eq. (1) for notched noise [Eq. (A5) in Appendix A] is mathematically equivalent, with the natural difference in symbols, to Equation A5 of [35]. Note that, for simplicity, Eq. (1) is only applicable to signals that are sinusoidal gratings with no spatial window or with fixed-size-spatial window [37,38]. B. Components of the Power Spectrum Model of Visual Masking In all simulations, the contrast thresholds (m0 ξi ) for detection of sinusoidal gratings were taken as the reciprocal of the contrast sensitivity (see Fig. 1(a) the following equation of [39]): CSu Au exp−au:

(2)

For the MTF of the channels, jHu; ξi j, we choose two forms [see Fig. 1(b)], one was symmetric on a linear frequency axis (asymmetrical on a logarithmic frequency axis) and the other was asymmetric on a linear frequency axis (symmetrical on a logarithmic frequency axis). From now



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0

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2.50 2 1.50 1 0.50 0 0

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Fig. 1. Components of the power spectrum model of visual masking. (a) CSF (A 337.323 and a 0.4455). (b) Symmetric and asymmetric MTFs. The peak spatial frequency is 4 c∕ deg and the bandwidth is 1.25 octaves. Black line, symmetric channel (σ 0.1148); gray line, asymmetric channel (α 0.3679). (c) Relationship between channel bandwidth (octaves) and its peak spatial frequency. Continuous line, exponential decreasing function (B 4.498 and b 0.6822); dashed line, constant function (B 0 and b 1.25). The parameters for the panels (a) and (c) are taken from Serrano–Pedraza and Sierra–Vázquez (2006, subject IS).

on we will use the term shape of channels corresponding to symmetric and asymmetric shape on a linear scale. As asymmetric MTF we used the popular log-Gaussian function [40] ( jHu; ξi j

h 2 i i exp − ln juj∕ξ ⇔u ≠ 0 2α2i ; 0 ⇔u 0

(3)

where ξi , ξi ≠ 0 is the peak spatial frequency of the channel i and αi , αi > 0 is an index of its spatial spread. Its relative bandwidth (full-width at half-height), Boct (ξi ), in octaves, is p 2 2 Boct ξi p αi : ln 2

(4)

As symmetric MTF we used the modulus of the Fourier transform of the 1D cosine Gabor function [41]: jHu; ξi j exp−2π 2 σ 2i u − ξi 2 exp−2π 2 σ 2i u ξi 2 ; (5) where ξi , ξi > 0 is the peak spatial frequency and σ i , σ i > 0 is an index of the spatial spread of the channel i. Its relative bandwidth (full-width at half-height), in octaves, is "p p# 2πσ i ξi ln 2 p : Boct ξi log2 p 2πσ i ξi − ln 2

(6)

We also adopted two forms for the relationship between Boct and the peak spatial frequency of the channel, an exponentially decreasing function of peak spatial frequency of the channel and a constant function [see Fig. 1(c)]. The equation for the relationship is Boct ξi b B exp−ξi ;

(7)

where b ≥ 0 and B ≥ 0. Depending on the values of the parameters b and B, Eq. (7) can show a decreasing relationship between channel bandwidth and peak spatial frequency. Its advantage over a logarithmic decreasing function [16,42] is that it prevents zero or negative values. On the other hand, if B 0, then Boct will be constant (in most of our simulations, we forced Boct to be constant).

According to previous masking results, we consider two relationships between channel bandwidth and peak spatial frequency [see Fig. 1(c)]. The first is a constant relationship in which the channels have the same bandwidth in octaves (constant bandwidth model), in this case B 0 [13]. The second one is a decreasing relationship where the bandwidth of the channels in octaves decreases with the peak frequency (variable bandwidth model) [16,17,42,43]. The solution of the integrals of the power spectrum model [Eq. (1)] for the two MTFs and the generic expressions for different types of masking noise can be seen in Appendix A. C. Model Parameters The model has five parameters (A, a, s, b, and B), two of them (A and a) determine the form of the contrast sensitivity function (CSF) [see Eq. (2) and Fig. 1(a)] and other two (b and B) the channel bandwidth variation with the peak spatial frequency. In order to make the simulations realistic, the values of A and a were chosen from psychophysical measurements of the CSF using Gabor patches with constant spatial window of 2.5 deg and mean luminance of 15 cd∕m2 (A 337.323 and a 0.4455), ([17], subject IS). The parameter s is the sensitivity parameter [16]. In most of the simulations the value of this parameter was s 0.3 and was chosen from previous masking experiments [37]. For the constant bandwidth model, we choose B 0 and b 1.25. For the variable bandwidth model, we chose the values obtained experimentally by [17], B 4.498 and b 0.6822. All parameters and their values used in the simulations can be seen in Table 1. Due to the similarity of the results obtained with the two different bandwidth assumptions, only the variable bandwidth model assumption will be used in the simulations where we study the effect of white noise level on contrast detection threshold. In all other simulations, the bandwidth of the MTF of the channels [see Eqs. (3) and (4)] was a constant value of 1.25 octaves [see Fig. 1(c), dashed line]. In the simulations, a total of 3200 channels were used with peak spatial frequencies within the range from 0.01 to 32 c∕deg in steps of 0.01 c∕deg [see examples of MTFs centered at spatial frequency of 4 c∕deg in Fig. 1(b)]. Once we know the peak spatial frequency, we compute the bandwidth of the channels according to Eq. (7), and finally the particular

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Table 1. Parameter Values of the Power Spectrum Model of Visual Masking A

a

s

b

B

Channel Peak

337.323 337.323

0.4455 0.4455

0.3 0.3

1.25 0.6822

0 4.498

ξi 0.01i; i 1; …; 3200 ξi 0.01i; i 1; …; 3200

Channel Model Constant bandwidth Variable bandwidth

channel spread for asymmetric channels was calculated using the equation p ln 2 αi p × Boct ξi ; 2 2 and for symmetric channels Boctξ p i 1 2 ln 2 1 × p × ; σ i Boctξ i 2 −1 π 2 ξi where ξi is the channel peak spatial frequency. D. Detection Models In order to compare the predictions of the power spectrum model of visual masking with and without taking into account off-frequency looking, we adopted, following [35] and [16], two detection models: the fixed-channel model and the best-channel model. 1. Fixed-Channel Detection Model In the fixed-channel detection model the spectral position of the detection channel remains fixed through the experiment. In particular, the target (sinusoidal grating) masked by noise is always detected by the channel centered on the spatial frequency of the signal (i.e., ξk u0 ) independently of the type of noise used. In this case Eq. (1) becomes 4 s

m2T u0 m20 u0

Z

∞

0

ρujHu; u0 j2 du

(following Patterson, 1974, 1976, with minor differences). 2. Best-Channel Detection Model In the best-channel detection model, the signal masked by noise is always detected by the channel with the maximum SNR at the output of the channel (that is, the “max (S/N)” assumption in [35]). Let r be the SNR function, that is, the ratio between the power of the signal at threshold at the output of each channel, P s u0 ; ξi ; mT , and the power of the noise at the output of this channel, P N ξi , (see [37], for a derivation of these two functions). At the output, the SNR for the channel with peak frequency ξi , for a sinusoidal grating of spatial frequency u0 and contrast, mT , ru0 ; ξi ; mT , is ru0 ; ξi ; mT

0

2

i

m2T u0 jHu0 ; ξi j2 2

2

sm20 ξi

R ∞ 0

ru0 ; ξk ; mT maxfru0 ; ξi ; mT g; i

where mT is the contrast detection threshold of the signal masked by noise. Taking the partial derivative ∂r∕∂ξi and equating to zero it is trivial to show that the particular value of mT ; mT u0 ≠ 0, has no computational effects on the estimation of the peak spatial frequency of the channel that maximizes the SNR, and therefore, we can ignore it without practical consequences. Actually, the problem of maximization r is theoretically equivalent to finding ξk , such as mT u0

∂ jHu0 ; ξi j2 R 0 ∞ 2 2 ∂ξi sm0 ξi 4 0 ρujHu; ξi j du ξi ξk

Thus, in order to compute the peak spatial frequency ξk of the channel that maximizes the SNR, we need to know the assumed channel MTF, the type of noise mask (characterized by means of its power spectrum), the power spectral density of the noise mask, the observer sensitivity, and the CSF. Alternatively, the same result can be achieved by finding the value of ξi that minimizes mT of Eq. (1), the procedure followed by Patterson and Nimmo-Smith (1980). Once the peak frequency ξk of the channel is known, then Eq. (1) will be used in order to obtain the squared-contrast detection threshold. Figure 2 shows a set of examples of the procedure used to obtain the peak frequency of the channel that maximizes the function r for different masks. The signal (sinusoidal grating) has a spatial frequency of 4 c∕ deg and the masks are ideal 1D bandpass noises with fixed bandwidth of 0.5 c∕ deg centered on a frequency within the range from 0.7 to 16 c∕ deg in half an octave steps. Figure 2(a) shows the power of the signal at the output of each channel (this function has been called the excitation pattern, [9]). Figure 2(b) shows the power of each bandpass noise (represented with different gray levels) at the output of each channel. Figure 2(c) shows the ratio of the functions in Fig. 2(a) and in Fig. 2(b). Each circle in the x–y plane corresponds to the peak frequency of the channel that maximizes the function r for each bandpass noise.

3. EFFECT OF THE TYPE OF NOISE ON CONTRAST THRESHOLD ESTIMATION

P s u0 ; ξi ; ms0 P N ξi

sm2 ξ

In the best-channel detection model, the channel involved in detection is the channel ξk , which maximizes ru0 ; ξi ; mT , that is

ρujHu; ξi j2 du

m2T u0 jHu0 ; ξi j2 R : 4 0∞ ρujHu; ξi j2 du

In the first simulation of a masking experiment we studied the effect of different types of noise masks on threshold estimation. For each noise mask, we compared the estimated contrast threshold for two detection models (fixed-channel and best-channel) and two MTFs (asymmetrical and symmetrical). The masks used in this simulation were white Gaussian noise



(b)

1000

uHF 2 2 sm0 (ξ i)+4N 0 |H(u,ξ i )| du uLF

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

x 10 -3

100

∫

|H(4, ξi)|2

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1123

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0.01

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0 1 2 3 4 5 6 7 8 9 10 11 12


0.5 1

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io

se rat

4 1. 2. 8

1

11 .3

0

se

i no

16

ss pa nd ba

8

of

5. 7

4

y nc ue eq

fr

2

er

t en C

2 P ea 3 k ch 4 an 5 ne lf 6 re qu 7 en 8 cy 9 (c /d 10 eg 1 1 )

12

2 1 0 0. 7

alize

Norm

1

(c)

i l-to-no sd igna

) eg /d (c

Fig. 2. Computational procedure used to obtain the peak frequency of the best channel (the channel that maximizes the SNR) using a fixed signal and a range of different masking bandpass noises of varying center frequency and constant bandwidth. (a) Power of the signal (grating of 4 c∕ deg) at the output of each channel (using the asymmetrical shape for the MTF of the channels). (b) Power of each bandpass noise at the output of each channel. The different bandpass noises are represented with different gray levels. (c) Ratio between the functions in (a) and (b). Each circle represents the peak frequency of the channel that maximizes the SNR. Note that when the center frequency of the bandpass noise does not match the spatial frequency of the signal there is off-frequency looking. The off-frequency looking is more extreme when the central frequency of the bandpass noise is closer to the signal’s spatial frequency.

(all-pass or broadband), low-pass noise, high-pass noise, notched noise, bandpass noise, and double bandpass noise. All simulations of masking experiments were performed using sinusoidal grating as signal and a 1D noise as mask. A. Method The signal was a grating of spatial frequency of 4 c∕deg. The bandwidth of the MTF of the channels [see Eqs. (3) and (4)] was a constant value, 1.25 octaves [see Fig. 1(c), dashed line]. The parameters of the masks used in this simulation study were as follows and appear in Table 2 and are used in the equations of Appendix A. – For the all-pass noise we used 12 power spectral density or noise levels, (N 0 0.002441 × 10−4 × 2n c∕deg−1 , n 0; …; 11). For the other noises, the noise level was always N 0 5 × 10−4 c∕deg−1 . – The low-pass noise was constructed using six cutoff frequencies ranging from 0.7071 to 4 c∕ deg in steps of 0.5 octaves. – For high-pass noise we used six cutoff frequencies ranging from 4 to 22.6274 c∕ deg in steps of 0.5 octaves. – For asymmetrical notched noise we used nine spectral notches sizes ranging from 0 to 4 octaves in steps of 0.5 octaves around 4 c∕ deg. When the shape of the notch was

symmetrical we used eight sizes ranging from 0 to 3.90 octaves; the notch shapes were symmetrical around 4 c∕deg and the gap increased in steps of 1 c∕deg. When the channel was asymmetrical, the notch was asymmetrical around the signal frequency and vice versa. – For the bandpass noise we used 10 center frequencies ranging from 0.7071 c∕deg to 16 c∕ deg in steps of 0.5 octaves; the width of the band was fixed at 0.5 c∕deg and the shape was symmetrical around the center frequency of the band. – For the asymmetrical double bandpass noise we used two flanking noise bands separated by eight different spectral gaps, ranging from 0.5 to 4 octaves in steps of 0.5 octaves. The width of each band was 0.5 c∕deg and each band was symmetrical around the center frequency of the bandpass noise. When the shape of the double bandpass noise was symmetrical, we used two flanking noise bands separated by seven different spectral gaps, ranging from 0.36 to 3.90 octaves. The flanking noises were symmetrical around 4 c∕deg and the gap increased in steps of 1 c∕deg. As notched noise, the gap of the flanking noise bands was asymmetrical around the signal frequency when the channel was asymmetrical and vice versa. [Note that although the maximum limit of the range is 32 c∕deg, higher frequencies than 20 c∕deg are invisible for the subject in our simulations according to the CSF used, see Fig. 1(a)].

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Table 2. Noise Types and Parameter Values (See Appendix A) Noise Masker

Power Level (N 0 ) −5

n

Spatial-Frequency Characteristics of the Noise −1

All-pass

N 0 0.02441 × 10

Low-pass

N 0 0.5 × 10−5 c∕deg−1

ulo 0 uhi 0.5 × 2nΔu , Δu 0.5 octave, n 1; …; 6

High-pass

N 0 0.5 × 10−5 c∕deg−1

ulo 4 × 2n−1Δu , Δu 0.5 octave, n 1; …; 6 uhi 32

Notched

N 0 0.5 × 10−5 c∕deg−1

u0 4 Asymmetrical Spectral notch size (octaves), Soct n − 1Δo, Δo 0.5, n 1; …; 9 ulo u0 × 2−Soct∕2 uhi u0 × 2Soct∕2 Symmetrical ulo u0 − n − 1Δu∕2, uhi u0 n − 1Δu∕2, Δu 1 c∕ deg, n 1; …; 8

Bandpass

N 0 0.5 × 10−5 c∕deg−1

uC 0.5 × 2nΔu , Δu 0.5 octave, n 1; …; 10 Bu 0.5 c∕deg

Double bandpass

N 0 0.5 × 10−5 c∕deg−1

u0 4 Bu 0.5 Asymmetrical Soct nΔo, Δo 0.5 octave, n 1; …; 8 uClo u0 × 2−Soct∕2 uChi u0 × 2Soct∕2 Symmetrical uClo u0 − nΔu∕2, uChi u0 nΔu∕2, Δu 1 c∕ deg, n 1; …; 7

× 2 c∕deg n 1; …; 12

B. Comparing the Estimated Thresholds for Two Detection Models (Fixed-Channel and Best-Channel) We wanted to assess both the advantage that would be conferred by off-frequency looking and the frequency difference that would give the greatest advantage. In order to do so, we compared the estimated thresholds for the two detection models (the fixed-channel and the best-channel). The difference was measured by comparing the coefficient of determination, R2 , so values closer to 1 indicate the thresholds are very similar up to a multiplicative constant. In the simulations of the best-channel detection model the thresholds are estimated using the channel that maximizes the SNR; therefore, for each noise condition a different channel is involved in detection. The difference in threshold between the models tells us whether off-frequency looking is likely to occur. The difference between the peak frequencies predicted by each model was measured by means of the root mean squared error (RMSE); RMSE values close to 0 indicate that the peak frequencies predicted by the two models are very similar. This measure is an indirect index of the amount (frequency difference) of off-frequency looking present in the particular experimental condition. C. Results and Discussion Figure 3 shows the results for asymmetrical channels and Fig. 4 for symmetrical channels for the two detection models (the fixed-channel and the best-channel). Figures 3 and 4 show the simulation results for (a) all-pass, (b) low-pass, (c) highpass, (d) notched, (e) bandpass, and (f) double bandpass noise. The upper panels show the estimated thresholds for the two

ulo 0 uhi 32

detection models as a function of the noise parameter (power spectral density, cutoff frequency, notch size, central frequency, and spectral gap). The lower panels show the peak spatial frequency of the channel involved in detection for both detection models. Figures 3 and 4 show the typical results found in visual masking experiments. The thresholds increase with the power density level of the white noise for both detection models [see Figs. 3(a) and 4(a), upper panels]. Thresholds also increase when the cutoff frequency of low-pass and highpass noises is close to the signal frequency [see Figs. 3(b), 3(c), 4(b), and 4(c), upper panels], when the center frequency of the bandpass noise is close to the signal’s spatial frequency [see Figs. 3(e) and 4(e), upper panels], and when both the notch size and the spectral gap of the double bandpass noise are small [see Figs. 3(d), 3(f), 4(d), and 4(f), upper panels]. In Figs. 3(b), 3(c), and 3(e) and Figs. 4(c) and 4(e) the estimated thresholds for the two detection models are different (the values of the coefficient of determination R2 are lower than 0.8). Figures 3(a)–3(f) (lower panels) and Figs. 4(a)–4(f) (lower panels) show the peak frequency of the channel mediating the detection in the two models. As described, Fig. 2 shows the procedure we used to obtain these peak frequencies; in particular the results correspond to those of Fig. 3(e). The graphs show that for low-pass, high-pass and bandpass noises, the peak frequencies predicted by the two detection models are very different. For example, in Fig. 3(b) (lower panel), when the cutoff frequency of the low-pass noise is 4 c∕deg, the best-channel model predicts that the peak frequency of the channel that detects the signal is approximately 7 c∕deg.



(b)

− 0.5

−1

− 1.5

− 1.5

−2

−2

−2

− 2.5

− 2.5

− 2.5

−3 0.001 0.01

−3 0.1

1

10

−3 0.5

8

2

4

2

8

All-pass

Low-pass

7

6

5

5

5

4

4

4

3

3

1 0.001 0.01

0.1

1

2

RMSE =1.2762

1

10

0.5

1

2

4

2

−1

− 1.5

− 1.5

− 1.5

−2

−2

−2

− 2.5

− 2.5

− 2.5

−3 2

3

4

5

8

Notch

7

2

4

8

0

16 32

Bandpass

6

5

5

5

4

4

4

3

3

3

2

2

0

1

2

3

4

Notch size (octaves)

RMSE =0.9073

1 5

2

3

4

5

0.5 1

2

Double bandpass

7

6

RMSE =0.2814

1

8

6

1

R 2 =0.9999

−3 0.5 1

8

7

32

Double bandpass

−1

1

16

− 0.5

−1

0

8

(f) R 2 =0.7762

Bandpass

−3

4

Cutoff frequency (c/deg)


− 0.5

R 2 =0.9971

Notch

32

RMSE =1.1425

1 8

(e)

− 0.5

16

3

2

RMSE =0.2739

8

High-pass

7

6

2

4

8

6

(d) Log test threshold (log10(m))

1

8

7

R 2 =0.6916

High-pass

−1

Power spectral density (c/deg)-1 (x10-4)


− 0.5

R 2 =0.8914

Low-pass

Best-channel Fixed-ch.

− 1.5

(c)

− 0.5

R 2 =0.9997

All-pass −1


Log test threshold (log10(m))

(a)

1125

4

8

16 32

Center frequency of bandpass noise (c/deg)

2

RMSE =0.1313

1 0

1

2

3

4

5

Spectral gap (octaves)

Fig. 3. Simulation results for asymmetrical channels. Circles show results for the best-channel detection model; squares show results for the fixedchannel detection model. (a) Upper panel: estimated thresholds (in log units) for a grating of 4 c∕ deg masked by a white noise as a function of the power spectral density. Lower panel: estimated peak spatial frequency of the channel mediating detection as a function of the power spectral density. (b) Upper panel: estimated thresholds masked by a low-pass noise as a function of the cutoff frequency. Lower panel: estimated peak frequency of the channel mediating detection as a function of the cutoff frequency. (c) Upper panel: estimated thresholds masked by a high-pass noise as a function of the cutoff frequency. Lower panel: estimated peak frequency of the channel mediating detection as a function of the cutoff frequency. (d) Upper panel: estimated thresholds masked by asymmetrical notched noise as a function of the spectral notch size. Lower panel: estimated peak frequency of the channel mediating detection as a function of the spectral notch size. (e) Upper panel: estimated thresholds masked by bandpass noise as a function of the center frequency of the noise. Lower panel: estimated peak frequency of the channel mediating detection as a function of the center frequency of the noise. (f) Upper panel: estimated thresholds masked by asymmetrical double bandpass noise as a function of the spectral gap. Lower panel: estimated peak frequency of the channel mediating detection as a function of the spectral gap. The horizontal gray line in the upper panels shows the threshold value for this grating in the absence of masking. The horizontal black line in the lower panels shows the peak spatial frequency of the channel centered on the signal frequency. R2 is the coefficient of determination between thresholds estimated by the two detection models. RMSE is the root mean squared error between peak channel frequencies estimated by the two detection models.

If there is a difference between the contrast thresholds estimated by the two detection models then we can confirm the presence of off-frequency looking because the channel that detects the signal estimated by the best-channel model is different from the spatial frequency of the signal. In general, the estimated thresholds are lower for the best-channel detection model than for the fixed-channel detection model. However, one could assume that if the contrast thresholds estimated by the two detection models (fixed-channel and best-channel) are identical or very similar then the noise that was used in the masking experiments prevents off-frequency looking. We will prove that this assumption it is not always correct. For example, in Fig. 4(b) (upper panel) the correlation between thresholds is high (R2 0.9748); this data could

suggest that there is no presence of off-frequency looking, but if we observe the lower panel of Fig. 4(b) (RMSE 1.1937) then we can confirm that in this case there is a clear presence of off-frequency looking. The interesting point in this case is that although we have off-frequency looking, thresholds are almost independent of this effect. As can be seen, none of the masks used in the simulations absolutely prevents the off-frequency looking. Comparing the RMSE values, we show, independently of the shape of the channels, that all-pass (from lower- to higher-power spectral density levels), notch, and double bandpass noises are the noises that strongly prevent off-frequency looking. However, low-pass, high-pass, and bandpass noises cannot prevent offfrequency looking. Similar results were found using different

1126



(b)

− 0.5

−1

− 1.5

− 1.5

−2

−2

−2

− 2.5

− 2.5

− 2.5

−3 0.001 0.01

−3 0.1

1

10

−3 0.5

All-pass

7

2

4

8

2

Low-pass

7

6

5

5

5

4

4

4

3

3

1 0.001 0.01

0.1

1

2

RMSE =1.1937

1

10

0.5

1

2

4

2


R 2 =0.7543

Bandpass

− 1.5

− 1.5

− 1.5

−2

−2

−2

− 2.5

− 2.5

− 2.5

−3 2

3

4

5

Notch

7

7

2

4

8

0

16 32

Bandpass

6

6

5

5

4

4

4

3

3

3

2

2

0

1

2

3

4


RMSE =0.8021

1 5

2

3

4

5

0.5 1

2

Double bandpass

7

5

RMSE =0.2372

1

8

6

1

R 2 =0.9941

−3 0.5 1

8

8

32

Double bandpass −1

1

16

− 0.5

−1

0

8


−1

−3

4

(f)

− 0.5

R 2 =0.9955

Notch

32

RMSE =0.7715

1 8

(e)

− 0.5

16

3

2

RMSE =0.2287

8

High-pass

7

6

2

4

8

6

(d) Log test threshold (log10(m))

1

8

8

R 2 =0.6640

High-pass

−1



− 0.5

R 2 =0.9748

Low-pass


− 1.5

(c)

− 0.5

R 2 =0.9998

All-pass −1



(a)

4

8

16 32

Center frequency of bandpass noise (c/deg)

2

RMSE =0.2296

1 0

1

2

3

4

5


Fig. 4. Simulation results for symmetrical channels. Circles show results for best-channel detection model; squares show results for fixed-channel detection model. (a) Upper panel: estimated contrast detection thresholds (in log units) for a grating of 4 c∕ deg masked by a white noise as a function of the power spectral density. Lower panel: estimated peak frequency of the channel mediating detection as a function of the power spectral density. (b) Upper panel: estimated thresholds masked by a low-pass noise as a function of the cutoff frequency. Lower panel: estimated peak frequency of the channel mediating detection as a function of the cutoff frequency. (c) Upper panel: estimated thresholds masked by a highpass noise as a function of the cutoff frequency. Lower panel: estimated peak frequency of the channel mediating detection as a function of the cutoff frequency. (d) Upper panel: estimated thresholds masked by symmetrical notched noise as a function of the spectral notch size. Lower panel: estimated peak frequency of the channel mediating detection as a function of the spectral notch size. (e) Upper panel: estimated thresholds masked by bandpass noise as a function of the center frequency of the noise. Lower panel: estimated peak frequency of the channel mediating detection as a function of the center frequency of the noise. (f) Upper panel: estimated thresholds masked by symmetrical double bandpass noise as a function of the spectral gap. Lower panel: estimated peak frequency of the channel mediating detection as a function of the spectral gap. The horizontal gray line in the upper panels shows the threshold value for this grating in the absence of masking. The horizontal black line in the lower panels shows the frequency of the channel centered on the signal frequency. R2 and RMSE the same as Fig. 3.

spatial frequencies for the signal (we used spatial frequencies within the range from 0.75 to 8 c∕deg) or using the assumption of variable relative bandwidth (results not shown).

4. EFFECT OF THE SHAPE OF NOTCHED NOISE AND DOUBLE BANDPASS NOISE ON CONTRAST THRESHOLD ESTIMATION We have shown that white noise, notched noise, and double bandpass noise prevent off-frequency looking. Our simulations show that white noise (all-pass) prevents off-frequency looking independently of the shape of the filter MTF [see Figs. 3(a) and 4(a)]. With regard to notched and double bandpass noises, in the simulations shown in Figs. 3(d), 3(e) and 4(d), 4(e) we matched the shape of the notch or the spectral gap (double bandpass noise) with the shape of the

channel (asymmetric or symmetric) so we do not know if these masks are effective at preventing off-frequency looking when the shape of the notch does not match the shape of the channel. In this section we study the case when the shape of the noise does not match the shape of the MTF of the channel. We simulate masking experiments using two types of shape (asymmetrical and symmetrical) for the notched noise and double bandpass noise, and using two types of shape (asymmetrical and symmetrical) for the MTF of the channels. A. Method In this study, the signal used was a grating of spatial frequency 4 c∕ deg. The bandwidth of the MTF of the channels [see Eqs. (3) and (4)] was a constant value of 1.25 octaves



panels of Fig. 5(a) show clearly that when the shape of the notch (symmetric) does not match the shape of the channel, the mask does not prevent off-frequency looking for notches wider than 1 octave. In fact, the right-most lower panel of Fig. 5(a) shows that the peak channel frequency mediating detection for the best-channel model moves away from the spatial frequency of the signal (4 c∕deg) as the gap size of the notch increases. Figure 5(b) shows the results when the MTF of the channel is symmetrical. The upper panel shows the results for the symmetric notch and the lower panel shows the results for the asymmetric notch. The upper panels of Fig. 5(b) show the same results obtained in Fig. 4(d). These results added to the previous result confirms that, independently of the shape

[see Fig. 1(c), dashed line]. The shapes of the two masks have been described previously in Section 2.A. B. Results and Discussion Figure 5 shows the results of the simulation of masking experiments using notched noise as a mask in four conditions. Figure 5(a) shows the results when the MTF of the filter is asymmetric [see also Fig. 1(b)]. The upper panel shows the results for the asymmetric notch and the lower panel shows the results for the symmetric notch. The upper panels of Fig. 5(a) show the same results obtained in Fig. 3(d). These results confirm that when the shape of the notch (asymmetric) matches the shape of the MTF of the channel, the mask prevents off-frequency looking. However, the lower

Notched noise

(a) Asymmetric channel

− 0.5

1 Asymmetric notch

0 0

2

4

6

8

10

1 Symmetric notch

0.75

− 2.5 −3 0

0

−3 8

3

4

R 2 =0.6340

10

Symmetric notch

2

3

4

4

6

8

10

1 Asymmetric notch

0.75

0


3

4

5

R 2 =0.9966

− 1.5

−3 10

2

−1

0 8

1

− 0.5

− 2.5

6

2

5

5

RMSE =0.7592 3

4

5


5

−3

0.25

4

1

4

6

6

− 2.5

−2

2

0

3

7

7

−2

0.50

0

2

8

R 2 =0.9955


2

1

8

5

− 1.5 Log test threshold (log10(m))

0

0

1 1

−1

0

RMSE =0.2814

1

2

− 0.5

0.25

2


1

0.50

3

3

0

Symmetric channel

0.75

4

4


(b)

5

− 1.5

− 2.5

6

2

−1

0.25

4

1

− 0.5

−2

2

5

−2

0.50

0

6 Peak channel frequency (c/deg)

0.25


7


0.50

Power gain

8

R 2 =0.9971

−1

0.75

Power gain

1127

4 3 2

RMSE =0.2372

1 0

1

2

0

1

2

3

4

5

8 7 6 5 4 3 2

RMSE =0.3484

1 0

1

2

3

4

5

3

4

5


Fig. 5. Effect of the shape of the notched noise on threshold estimation for different channel shapes. Circles show results for the best-channel model; squares show results for the fixed-channel model. Left panels show the MTF of the channel and an example of the notched noise used in the simulations, the arrow indicates the spatial frequency of the signal. Center panels, estimated contrast detection thresholds (in log units) for a grating of 4 c∕deg masked by notched noise as a function of notch size; right panels, peak channel frequency as a function of notch size. (a) Results for asymmetrical channels of constant bandwidth of 1.25 octaves. Upper panels, results for asymmetrical notched noise; left panel, sample of one asymmetrical channel with peak frequency of 4 c∕deg (black line) and one sample of the mask with a spectral gap of 1 octave (shaded area). Lower panels, results for symmetrical notched noise; left panel, sample of one asymmetrical channel with peak frequency of 4 c∕deg and one sample of the mask with a spectral gap of 2 c∕deg. (b) Results for symmetrical channels of constant bandwidth of 1.25 octaves. Upper panels, results for symmetrical notched noise; left panel, sample of one symmetrical channel centered on 4 c∕deg and one sample of the mask with a spectral gap of 2 c∕deg. Lower panels, results for asymmetrical notched noise; left panel, sample of one symmetrical channel centered on 4 c∕deg and one sample of the mask with a spectral gap of 1 octave. The horizontal gray line in the center panels shows the threshold value for this grating in the absence of masking. The horizontal black line in the right panels shows the frequency of the channel centered on the signal frequency. R2 is the coefficient of determination between thresholds. RMSE is the root mean squared error between peak channel frequencies.



Figure 6 shows the results of the simulation using double bandpass noise as a mask in four conditions. The structure of this figure is identical to that of Fig. 5. Figure 6(a) shows the results when the MTF of the channel has an asymmetric shape and Fig. 6(b) shows the results when it has a symmetric shape. The results are very similar to those described in Fig. 5. Note again that for asymmetric double bandpass noise and symmetric channels, the threshold for the two detection models are very similar although the estimated peak channel frequencies are different. Simulations for both types of mask using a decreasing relationship between bandwidth of the channel and its peak spatial frequency [see Fig. 1(c), black line] gave similar results (results not shown).

of the filter, if the shape of the notch matches the shape of the MTF of the channel, the mask will prevent off-frequency looking. However, as shown before, the lower panels show clearly that when the shape of the notch does not match the shape of the filter, the mask does not prevent off-frequency looking, in this case for widths of the notch between 2 and 3 octaves [see Fig. 5(b), lower panel]. Note that in this case the thresholds for the two detection models are very similar although the estimated peak channel frequencies are different. If notched noise is going to be used in a masking experiment and we do not know the shape of the channel we recommend using notched noise with an asymmetric shape. Our results show that this asymmetric shape prevents off-frequency looking more strongly than the symmetric shape.

Double bandpass noise

(a) Asymmetric channel

− 0.5

1

0 0

2

4

6

8

10

1 Symmetric double bandpass

0.75

− 2.5 −3 0

− 2.5

0

−3 6

8

3

5

R 2 =0.7031

10

Symmetric double bandpass

2

3

4

2

4

6

8

10

1 Asymmetric double bandpass

0.75


2

3

4

5

R 2 =0.9864

−1 − 1.5

−3 10

1

− 0.5

0 8

2

5

5

RMSE =0.6599 3

4

5


5

0

− 2.5

6

1

4

6

6

−3

0.25

4

0

3

7

7

− 2.5

−2

2

2

8

R 2 =0.9941

−2

0.50

0

1

8

5


0

0

1 1

−1

0

RMSE =0.1313

1

2

− 0.5

0.25

2


1

0.50

3

3

0

Symmetric channel

0.75

4

4


(b)

4

− 1.5

0.25

4

2

−1

−2

2

1

− 0.5

0.50

0

5

−2


Power gain

0.25

6


0.50


7

−1 Asymmetric double bandpass

0.75

Power gain

8

R 2 =0.9999


1128

4 3 2

RMSE =0.2296

1 0

1

2

0

1

2

3

4

5

8 7 6 5 4 3 2

RMSE =0.4726

1 0

1

2

3

4

5

3

4

5


Fig. 6. Effect of the shape of the double bandpass noise on threshold estimation depending of the channel shape. Circles show results for bestchannel model; squares show results for fixed-channel model. Left panels show the MTF of the channel and an example of the notched noise used in the simulations, the arrow indicates the spatial frequency of the signal. Center panels, estimated contrast detection thresholds (in log units) for a grating of 4 c∕deg masked by double bandpass noise as a function of the spectral gap; right panels, peak channel frequency as a function of the spectral gap. (a) Results for asymmetrical channels of constant bandwidth of 1.25 octaves. Upper panels, results for asymmetrical spectral gap; left panel, sample of one asymmetrical channel with peak frequency of 4 c∕ deg (black line) and one sample of the mask with a spectral gap of 1 octave (shaded area). Lower panels, results for symmetrical spectral gap; left panel, sample of one asymmetrical channel with peak frequency of 4 c∕deg and one sample of the mask with a spectral gap of 2 c∕deg. (b) Results for symmetrical channels of constant bandwidth of 1.25 octaves. Upper panels, results for symmetrical spectral gap; left panel, sample of one symmetrical channel centered on 4 c∕deg and one sample of the mask with a spectral gap of 2 c∕deg. Lower panels, results for asymmetrical spectral gap; left panel, sample of one symmetrical channel centered on 4 c∕deg and one sample of the mask with a spectral gap of 1 octave. The horizontal gray line in the center panels shows the threshold value for this grating in the absence of masking. The horizontal black line in the right panels shows the frequency of the channel centered on the signal frequency. R2 is the coefficient of determination between thresholds. RMSE is the root mean squared error between peak channel frequencies.



5. EFFECT OF THE POWER SPECTRUM LEVEL OF WHITE NOISE AND THE VARIATION OF CHANNEL BANDWIDTH ON THE CONTRAST SENSITIVITY FUNCTION

B. Results and Discussion Figure 7 shows the results of the simulations. Figure 7(a) shows the results for a constant relative channel bandwidth, and Fig. 7(b) shows the results for a variable relative bandwidth (octave bandwidth decreasing with increasing peak spatial frequency). Upper panels show the bandwidth in octaves as a function of the peak channel frequency. Lower panels show the estimated contrast thresholds for six spatial frequencies and five power density levels. Figures 7(a) and 7(b) (lower panels) show that the contrast detection thresholds increase with the power density level of the noise. The particular shape of the masking functions in Figs. 7(a) and 7(b) depends on the relationship between the bandwidth of the channels and their peak spatial frequency. With constant channel bandwidth, the threshold increases with the spatial frequency of the signal for each level of noise. In contrast, with decreasing channel bandwidth the thresholds for each level of noise are almost independent of the spatial frequency of the signal (the masking curves become almost flat). This latter result is the typical empirical result obtained when measuring detection thresholds of sinusoidal gratings masked with white noise [15,17,23,44,45]. Thus, the model predicts that the bandwidths of the visual channels decrease with increasing peak spatial frequency, a result that is in agreement with previous empirical data [16,17,42,43]. In order to study off-frequency looking with white noise and for each type of bandwidth relationship we performed comparisons between the estimated thresholds of two detection models (fixed-channel and best-channel) fixing the spatial frequency of the signal at different density levels (thresholds

We have shown that white noise (all-pass) prevents offfrequency looking independently of the shape of the channel [see Figs. 3(a) and 4(a)]. Here, we will study whether the white noise also prevents off-frequency looking for a broad range of spatial frequencies and for different noise levels. At the same time, we will also show the effect of the relationship between channel bandwidth and peak channel frequency on CSF. A. Method In this simulation study, the signal used was a grating with six different spatial frequencies ranging from 0.25 to 8 c∕deg in steps of 1 octave. The mask was white noise with five power spectral density levels, (N 0 0.00195 × 10−3 × 22n c∕deg−1 , n 0; …; 4). We will show the results for an asymmetric shape of the MTF of the channels, although we will discuss the results for both symmetrical and asymmetrical shapes. In the simulations, we adopt two previously described forms for the relationship between Boct and the peak spatial frequency of the MTF of the channels [see Eq. (5) and Fig. 1(c)], one has a constant bandwidth of 1.25 octaves and the other has an exponentially decreasing shape [see parameters in Table 1 and Fig. 1(c)]. We compare the estimated thresholds and the peak channel frequencies for two detection models (fixed-channel and bestchannel).

(a) Bandwidth (octaves)

1129

(b) 3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5 0

0 0

2

4

6

8

0

10 12 14 16 18 20

2

4

6

8

10 12 14 16 18 20



− 0.5

− 0.5

Best-channel Fixed-ch. −1

−1

− 1.5

− 1.5

−2

−2

− 2.5

− 2.5 0.1

1

10

20

0.1

1

10

20


Fig. 7. Effect of white noise level and bandwidth of channels on the CSF. (a) Simulation results with constant bandwidth for six spatial frequencies and five power spectral levels. Upper panel, bandwidth in octaves as a function of the peak channel frequency; lower panel, estimated contrast detection thresholds (in log units) of the test as a function of the spatial frequency parameterized by the noise level of the white noise. Circles, results for the best-channel detection model; squares, results for the fixed-channel detection model. The thick black line represents the inverse CSF (detection thresholds in absence of masking). Note that thresholds obtained with the lowest power density level are closer to this curve. (b) Simulation results with decreasing bandwidth for six spatial frequencies and five power spectral levels. Upper panel, bandwidth in octaves as a function of the peak channel frequency; lower panel, log-thresholds of the test as a function of the spatial frequency parameterized by the noise level of the white noise. The thick black line represents the inverse function of the CSF.

1130



Table 3. Comparison Between Power Density Levelsa Spatial Frequency of the Signal (c/deg)

R2

(constant bandwidth) R2 (decreasing bandwidth) RMSE (constant bandwidth) RMSE (decreasing bandwidth)

0.25

0.5

1

2

4

8

0.995 0.971 0.017 0.109

0.999 0.998 0.026 0.148

0.999 0.999 0.049 0.034

0.999 0.999 0.120 0.012

0.999 0.999 0.266 0.059

0.996 0.998 0.733 0.303

a The values of R2 (for constant or decreasing bandwidth) measure the difference between the thresholds estimated by fixed-channel and best-channel detection models (Fig. 7) for six spatial frequencies of the signal and for five power levels of the white noise. The values of RMSE measure the difference between the peak frequencies mediating detection estimated by fixed-channel and best-channel models. Values of R2 closer to 1 indicate that the thresholds predicted by both models are very similar. Values of RMSE closer to 0 indicate that the peak frequencies of the channels mediating detection predicted by the two detection models are very similar.

were compared calculating the coefficient of determination R2 ). We also performed comparisons between the peak frequencies of the channels mediating detection for both detection models (we calculate the value of the RMSE). Table 3 shows the results of these comparisons. The value R2 shows that the thresholds are very similar for both detection models and both bandwidth-peak channel frequency relationship. The RMSE shows that the white noise does not prevent offfrequency looking except when the spatial frequency is 8 c∕deg and the channel bandwidth is constant, although the thresholds for the two detection models are very similar (R2 0.996).

6. FITTING THE POWER-MASKING MODEL TO VISUAL MASKING DATA In this section we will show the results of visual masking experiments for two subjects (ISP and GB; GB was unaware of the purpose of the study) and the fitting of the power spectrum model of visual masking to the data. The task of the masking experiments was to detect a horizontal Gabor patch of 1 c∕deg with a spatial standard deviation of 2.5 deg masked by six different types of 1D noise (all-pass, low-pass, highpass, notched, bandpass, and double bandpass noise). A. Stimuli and Equipment The stimuli (images of 512 × 512 pixels, subtending 8° × 8°, and mean luminance of 15 cd∕m2 ) were presented on a gamma corrected 1900 monitor (Eizo Flexcan T765, Eizo Corp., Japan) in gray scale (frame rate of 120 Hz) using a VSG2/3F Issue 4a graphics card (Cambridge Research Systems Ltd., UK) that provides 15 bit gray scale resolution. The images of the 1D noise masks were constructed in the Fourier domain (see procedure for 1D white noise in Serrano–Pedraza and Sierra–Vázquez, 2006). First, we generated the amplitude spectrum for the particular noise mask (given the characteristics of the apparatus, the noises did not have energy in spatial frequencies lower than 0.125 c∕deg and higher than 16 c∕ deg); second, we generated the phase spectrum where the phase values were random variables uniformly distributed between −π; π; and third, we transformed the Fourier spectra into the two-dimensional spatial domain. The parameters of the masks used in the experiments were as follows. For the all-pass noise we used 10 power spectral density or noise levels (N 0 0.0097656 × 10−4 × 2n c∕deg−1 , n 0; …; 9). For the other noises, the noise level was always N 0 5 × 10−4 c∕deg−1 . The low-pass noise was constructed using five cutoff spatial frequencies ranging from 0.25 to 1 c∕deg in steps of 0.5 octaves. For high-pass noise we used

six cutoff frequencies ranging from 1 to 5.65 c∕deg in steps of 0.5 octaves. For the notched noise we used six spectral asymmetrical notches sizes [see example in Fig. 5(a), upper panel] ranging from 0 to 5 octaves in steps of 1 octave around 1 c∕deg. For the bandpass noise we used 10 center frequencies ranging from 0.25 c∕deg to 5.65 c∕deg in steps of 0.5 octaves; the size of the band was fixed at 0.5 c∕deg and the shape was symmetrical around the center frequency of the band. For the double bandpass noise we used two flanking noise bands separated by four different asymmetrical spectral gaps around 1 c∕deg [see example in Fig. 6(a), upper panel], ranging from 1 to 4 octaves in steps of 1 octave (the width of each band was 0.5 c∕deg). B. Procedure The subjects sat in a dark room and observed the screen from 105 cm, fixing the distance by means of a chin rest. The signal and the mask were presented using the frame interleaving method [16,38,45]. The signal was presented in odd frames and the masking noise in even frames, so we could change the contrast (using the look-up table) of the signal independently of the contrast of the masking noise. We used a two-interval forced choice task and a Gaussian envelope for the temporal presentation. The duration (2σ t ) of the stimulus was 100 ms (the Gaussian envelope was truncated to obtain an overall duration of 500 ms) and an interstimulus interval was 500 ms (a fixation cross was presented during these intervals). In the masking experiments, the signal was presented in one of the intervals and a different stochastic noise was used in the two presentation intervals. Thus, the task was to indicate in which interval the signal was presented. In order to measure contrast detection thresholds we used an adaptive Bayesian staircase similar to the method called ZEST [46], the specific details can be seen in [38]. Each threshold shown in Fig. 8 corresponds to the mean of three thresholds estimated using staircases of 70 trials. C. Fitting the Power Spectrum Model Figure 8 shows the results of the masking experiments for two subjects. Rows (a) and (b) show the data of subject ISP, and rows (c) and (d) show the data of subject GB. Each column shows the results for a different type of mask, from left to right: all-pass, low- and high-pass, notched, bandpass, and double bandpass noise. Each panel shows the contrast detection threshold (in logarithmic units) for a Gabor patch of spatial frequency of 1 c∕deg as a function of the masking noise parameter. The results of both subjects are very similar to



(a)


All-pass

− 0.5 −1 − 1.5 −2 − 2.5

Low-pass & High-pass

Best-channel Data R 2 =0.9462 b =3.3498 s =0.5052

1

Double band-pass

− 0.5

− 0.5

− 0.5

−1

−1

−1

−1

− 1.5

− 1.5

− 1.5

− 1.5

−2

−2

−2

−2

− 2.5

ISP

−3 0.1

Band-pass

− 0.5

− 2.5

ISP

−3 0.001 0.01

Notch

0.1 0.25 0.5 1

10

− 2.5

ISP

2

4

8 16

− 2.5

ISP

−3

−3 0

1

2

3

4

ISP

−3 0.1 0.25 0.5 1

5

1131

2

4

8

0

1

2

3

4

5

2

3

4

5

2

3

4

5

2

3

4

5


(b) − 0.5 −1 − 1.5 −2 − 2.5

Fixed-channel Data R 2 =0.8866 b =2.8891 s =0.7432

− 0.5

− 0.5

− 0.5

−1

−1

−1

−1

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−3 0.001 0.01

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ISP

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5

ISP

−3 2

4

8

0

1


(c) − 0.5 −1 − 1.5 −2 − 2.5

Best-channel Data R 2 =0.8017 b =3.6259 s =0.3429

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GB

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8 16

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GB

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5

GB

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8

0

1


(d) − 0.5 −1 − 1.5 −2 − 2.5

Fixed-channel Data R 2 =0.9046 b =2.9248 s =0.4194

− 0.5

− 0.5

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−1

−1

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GB

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5

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8

Center frequency of BP noise(c/deg)

0

1


Fig. 8. Results from visual masking experiments for two subjects (ISP and GB). Each column shows the contrast detection thresholds in log units (circles; in the low-pass and high-pass column, the squares show thresholds for high-pass noise) for a Gabor patch of 1 c∕deg masked by a particular type of noise (all-pass, low-pass and high-pass, notched, bandpass, and double bandpass noise). In each panel there is a sketch of the amplitude spectrum of the noise used in the experiment. (a), (b) Both rows show the contrast thresholds (mean S:D:) of the subject ISP as a function of the power spectral density (all-pass), cutoff frequency (low- and high-pass), notch size, center spatial frequency (bandpass) and spectral gap (double bandpass). The empirical thresholds in the two rows are the same. (c), (d) These rows show the contrast thresholds (mean S:D:) of the subject GB. (a), (c) In these two rows, the black line represents the fitted power spectrum masking model assuming asymmetric channel MTF [see Eq. (3)], constant bandwidth, and best-channel detection model. The left-most panel from the left shows the parameters estimated: b (bandwidth in octaves) and s (sensitivity). (b), (d) In these two rows, the black line is the fitted power spectrum masking model assuming asymmetric channel MTF, constant bandwidth, and fixed-channel detection model. The horizontal gray line in all panels shows the threshold for the Gabor patch of 1 c∕deg without masking. The value of R2 shown in the left panel of each row is the coefficient of determination between all masking thresholds from the five panels and the model predictions.

the simulated results shown in Fig. 3, although those of Fig. 3 were for a signal of spatial frequency of 4 c∕deg. The black lines in each panel correspond to the fitted power spectrum masking model. For the fitting we have assumed two shapes for the MTF of the channels (in Fig. 8 we only show the results for the asymmetric shape because the model fitted the data much better). We also assumed that the bandwidths of all channels were the same [see Fig. 1(c), dashed lines; in Eq. (7), B 0]. For the CSF we used Eq. (2) with parameters A 337.323 and a 0.4455 for subject ISP, and A 249.76 and a 0.3424 for subject GB (the parameters of the CSFs belong to the same subjects in the same experimental conditions [17]); and we assumed two detection models (fixed- and best-channel). Thus, the model had only

two free parameters, b, the bandwidth of the channels in octaves, and s, the sensitivity parameter. We fitted the power spectrum model to all masking data (41 thresholds) obtained with the six types of noise. In rows (a) and (c) of Fig. 8 we fitted the model assuming detection by the “best” channel. In rows (b) and (d) we fitted the model assuming detection by a “fixed” channel centered on the spatial frequency of the signal. The fitting was performed using the equations of Appendix A and the “best” channel was obtained according to the procedure explained in Section 1.4 (see Fig. 2). The values of parameters b and s were estimated using a least-squares fitting procedure. The sum of the squared errors between the empirical log-thresholds and the model log-thresholds was minimized using the Matlab routine

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“fminsearch,” which uses the Nelder–Mead simplex search method [47]. The coefficient of determination (R2 ) between all empirical masking thresholds from the five panels and the model predictions is presented in the first panel of each row. Figure 8 shows, for different detection models, that the power spectrum model of visual masking fits empirical data very well even using only two free parameters (bandwidth and sensitivity). According to the coefficient of determination, for the subject ISP [see Figs. 8(a) and 8(b)], the asymmetric channels with the best-channel detection model (R2 0.946) fits better than the fixed-channel model (R2 0.886); however, for the subject GB (see Figs. 8(c) and 8(d)], the fixed-channel detection model (R2 0.905) fits better than the best-channel model (R2 0.802). This difference in the strategies of detection adopted by the two subjects can be explained by the difference in the experience in psychophysical experiments [11]. Although subject GB has experience in psychophysical experiments, he does not have experience in detection experiments. Pelli (1981) showed that for low- and high-pass noise the thresholds reduced with practice, however this did not happen for broadband noise (all-pass). He suggested that off-frequency looking could cause this difference. The bandwidths (FWHM), obtained from the best-fitted model, of the MTF of the channels for both subjects were 3.3 octaves (ISP) and 2.9 octaves (GB). These bandwidths are similar to those obtained by [16] using a similar model although for us the bandwidths obtained using the best-channel detection model were always bigger than using the fixed-channel model, similar to the auditory modeling [26].

7. GENERAL DISCUSSION AND PRACTICAL RECOMMENDATIONS This paper presents a simulation study in which the power spectrum model of visual masking was tested assuming a visual observer (with different MTFs, channel bandwidths, and a particular CSF), two detection models (fixed- and bestchannel), and different types of noise masks in order to establish the conditions in which off-frequency looking takes place. Simulations confirm that (1) high-pass, low-pass, and bandpass noises do not prevent off-frequency looking, a result in agreement with empirical results [11,12,16,30]; (2) white noise satisfactorily prevents off-frequency looking independently of the shape and bandwidth of the MTF of the channel; and (3) notched and double bandpass noises fail to prevent off-frequency looking unless the shape of the mask matches the symmetrical or asymmetrical shape of the visual channel involved in detection. Interestingly, we obtained a surprising result for some noise masks: even if off-frequency looking existed, predicted thresholds were the same as they would have been without it. The main question that these results pose is this: can we make a masking experiment using notched or double bandpass noise in a way that prevents off-frequency looking [12–14,31]? Our simulation results show that we can only be sure that off-frequency looking will not affect the results if we know a priori the shape of the MTF of the channel and not if we only know whether or not it is bandpass. In order to use notched or double bandpass noise to prevent off-frequency looking, we need to know approximately the


shape of the MTF of the channels. To estimate the shape of the channel MTF using masking, we recommend the use of bandpass noise, the results plotted in Figs. 3(e) and 4(e) show that with this noise it is relatively easy to find out what could be the shape of the channel (even in presence of off-frequency looking). If we are not sure about the shape, our results suggest using an asymmetric shape for the notch and for the spectral gap of the double bandpass noise. This asymmetric shape showed less off-frequency looking (see Figs. 5 and 6). Our results also show that white noise prevents offfrequency looking independently of the shape of the channels and their bandwidth and that white noise could be a useful tool if we want to obtain the relationship between the bandwidth of the channels and their peak spatial frequency [13,17], but it is not useful for estimating the exact value of the bandwidth because that depends on the sensitivity value s [17]. In some cases, depending on the mask and filter shape, our results show that off-frequency looking has a small or no effect on contrast detection threshold [see Figs. 4(b), 5(b), and 6(b)]. This means that although the signal is detected by a channel with peak frequency different from the frequency of the signal, the estimated threshold does not change. This result is clear when the mask is a low-pass noise and the assumed channel is symmetrical [see Fig. 4(b)]. This result is also evident for symmetrical channels when the mask is asymmetric notched noise [see Fig. 5(b)] or asymmetric double bandpass noise [see Fig. 6(b)] or bandpass noise [see Fig. 4(e)]. Thus, the use of the contrast detection thresholds to test the existence of off-frequency looking can lead to wrong conclusions. In general a reduction in contrast detection thresholds in different sessions could be due to off-frequency looking [11]; however, if thresholds do not change then we can have two explanations, the first is that the signal is always detected by the same channel, the second is that the signal is detected by the channel centered on a spatial frequency different from the signal but it has no effect upon the threshold. Thus, we recommend that whenever the power spectrum masking model is used to interpret data from masking experiments both fixed-channel and bestchannel detection models should be used, regardless of the type of masking noise. Finally, we have shown empirical masking data for six different types of 1D noise (all-pass, low-pass, high-pass, notched, bandpass, and double bandpass noise) and examples of the fitting of the power spectrum model of visual masking for fixed- and best-channel detection models. Our results show that under reasonable assumptions and using only two free parameters the model provides an excellent fit (R2 0.946 for subject ISP and R2 0.905 for GB subject) to 41 contrast detection thresholds obtained from masking experiments (see Fig. 8).

APPENDIX A In this appendix we will show the solution of the definite integrals for the two proposed MTFs of the visual channels. These integrals are useful for making simulations of masking experiments using the power spectrum model. We will also show the particular expressions of the integrals for each type



of noise (white, low-pass, high-pass, notched, bandpass, and double bandpass noise). A. Definite Integrals for the Two Proposed MTFs of Visual Channels 1. Asymmetric Channel The expression of this MTF can be seen in Eq. (3). We solved the integrals within the intervals ulo (low spatial frequency) and uhi (high spatial frequency). This general expression is useful in order to solve the integral of the model when we multiply the MTF of the channel by the power spectrum ρ of the mask [see Eq. (1)]: Z

uhi ulo

2 2 p αi π α i ξi ln u∕ξi du exp − exp 2 4 α2i ulo 2 lnuhi ∕ξi −αi ∕2 lnulo ∕ξi −α2i ∕2 −erf ; (A1) × erf αi αi

Z

uhi

where u > 0, ξi > 0 (ξi corresponds to the peak of the MTF), uhi > ulo ≥ 0, αi > 0, (index of the spatial spread of the MTF of the channel i), and erfx is the error function defined as follows: 2 erfx p π

Z

x 0

exp−t2 dt

uhi

ulo

uhi

ulo

exp −4π 2 σ 2i u − ξi 2 exp −4π 2 σ 2i u ξi 2

2 exp −4π 2 σ 2i u2 ξ2i du 1 p exp −4π 2 σ 2i ξ2i × erf2πσ i uhi − erf2πσ i ulo 2 πσ 1 erf2πσ i uhi ξi − erf2πσ i ulo ξi 2

1 (A2) erf2πσ i uhi − ξi − erf2πσ i ulo − ξi ; 2 where uhi > ulo ≥ 0; σ i > 0, is an index of the spatial spread of the channel i, and erfx is the error function defined above. 3. Power Spectrum Model Expressions for Each Mask Used in the Simulations The general expression of the power spectrum model of visual masking can be seen in Eq. (1). In this section we will show the particular expressions for each type of noise. Using (A1) or (A2) for each particular expression, the solution for each type of noise follows below. White noise (all-pass). The noise power density is constant, therefore ρu N 0 , and Eq. (1) is as follows: m2T u0 ; uhi ; N 0

R m20 ξk 4Ns 0 0uhi jHu; ξk j2 du : jHu0 ; ξk j2

Ru

hi

ulo

jHu; ξk j2 du

jHu0 ; ξk j2

;

(A4)

where uhi 32 c∕ deg and ulo corresponds to the cutoff frequency of the high-pass noise. In Figs. 3(b), 3(c), 4(b), and 4(c) log contrast threshold is plotted as a function of the cutoff spatial frequency (uhi for low-pass noise or ulo for high-pass noise) with fixed u0 and N 0 as parameters. Low-pass: ulo 0, uhi 0.5 × 2nΔu , Δu 0.5 octave, n 1; …; 6. High-pass: ulo 4 × 2n−1Δu , Δu 0.5 octave, n 1; …; 6, uhi 32. Notched noise. The notched noise is constructed as a sum of two noises, one is a low-pass noise and the other is a highpass noise. The equation used in the simulations is as follows:

(A5)

jHu; ξi j2 du Z

m20 ξk 4Ns 0

m2T u0 ; ulo ; uhi ; N 0 i hR R m20 ξk 4Ns 0 0ulo jHu; ξk j2 du u32hi jHu; ξk j2 du ; jHu0 ; ξk j2

2. Symmetric Channel The expression of this MTF can be seen in Eq. (4): Z

We used uhi 32 c∕ deg. In Figs. 3(a) and 4(a) log contrast threshold is plotted as a function of power level N 0 with fixed u0 and uhi as parameters, N 0 0.02441 × 10−5 × 2n c∕deg−1 , n 1; …; 12, and in Fig. 7 it is plotted as a function of u0 , but parameterized by N 0 , N 0 0.00195 × 10−3 × 22n c∕deg−1 , n 0; …; 4. Low-pass and high-pass noise. For low-pass noise the equation is the same as for the white noise but the upper limit of the integral (uhi ) corresponds to the cutoff frequency of the low-pass noise. For high-pass noise the equation is as follows: m2T u0 ; ulo ; N 0

jHu;ξi j2 du

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(A3)

where ulo is the cutoff frequency of the low-pass component and uhi is the cutoff frequency of the high-pass component. Let S oct be the spectral notch size in octaves (S oct log2 uhi ∕ulo ). For asymmetric notched noises [Figs. 3(d), 5(a) (upper), 5(b) (lower)] ulo u0 2−Soct ∕2 , uhi u0 2Soct ∕2 , S oct n − 1Δo, Δo 0.5 octaves, n 1; …; 9. For symmetric notched noises [Figs. 4(d), 5(a) (lower), and 5(b) (upper)] ulo u0 − n − 1Δu∕2, uhi u0 n − 1Δu∕2, Δu 1 c∕deg, n 1; …; 8. In all of these panels log contrast threshold is plotted as a function of spectral notch size S oct , in octaves, with fixed u0 and N 0 as parameters. Bandpass noise. In the simulations using bandpass noise, Eq. (1) becomes in the next equation: R u Bu m20 ξk 4Ns 0 u C−Bu2 jHu; ξk j2 du C 2 2 mT u0 ; Bu ; uC ; N 0 ; (A6) jHu0 ; ξk j2 where Bu , in c/deg, is the absolute bandwidth of the bandpass noise; in all simulations the parameter Bu had the value of 0.5 c∕ deg. The variable uC is the central frequency of the bandpass noise, uc 0.5 × 2nΔu , with Δu 0.5 c∕deg, n 1; …; 10. In Figs. 3(e) and 4(e) log contrast threshold is plotted as a function of center frequency uC with fixed u0 , Bu , and N 0 as parameters. Double bandpass noise. The double bandpass noise is constructed as a sum of two bandpass noises centered on different spatial frequencies. For the double bandpass noise, Eq. (1) becomes in the next equation:

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m2T u0 ;Bu ;uClo ;uChi ;N 0 i hR R u Bu u Bu m20 ξk 4Ns 0 u Clo−Bu2 jHu;ξk j2 du u Chi−Bu2 jHu;ξk j2 du Clo 2 Chi 2 ; jHu0 ;ξk j2 (A7) where Bu is the absolute bandwidth of the bandpass noises (Bu 0.5 c∕ deg). The variable uClo is the central frequency of the lower bandpass noise, and uChi is the central frequency of the higher bandpass noise. Let S oct be the spectral gap size in octaves (S oct log2 uChi ∕uClo ). For asymmetrical double bandpass noises [Fig. 3(f), 6(a) (upper), and 6(b) (lower)] uClo u0 2−Soct ∕2 , uChi u0 2Soct ∕2 ,S oct nΔo, Δo 0.5 octaves, n 1; …; 8. For symmetrical double bandpass noises [Figs. 4(f), 6(a) (lower), and 6(b) (upper)] uClo u0 − nΔu∕2, uhi u0 nΔu∕2, Δu 1 c∕deg, n 1; …; 7. In all of these panels, log contrast threshold is plotted as a function of spectral gap size S oct , in octaves, with fixed u0 , Bu , and N 0 as parameters.

ACKNOWLEDGMENTS Supported by grant PSI2011-24491 from Ministerio de Economía y Competitividad (Spain) to I. Serrano-Pedraza.

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